Mathematical Explanation by Law
forthcoming in the British Journal for the Philosophy of Science

Sam Baron

Abstract

Call an explanation in which a non-mathematical fact is explained — in part or in whole
— by mathematical facts: an extra-mathematical explanation. Such explanations have
attracted a great deal of interest recently in arguments over mathematical realism. In this
paper, a theory of extra-mathematical explanation is developed. The theory is modeled on
a deductive-nomological theory of scientific explanation. A basic DN account of
extra-mathematical explanation is proposed and then redeveloped in the light of two
difficulties that the basic theory faces. The final view appeals to relevance logic and uses
resources in information theory to understand the explanatory relationship between
mathematical and physical facts.

1 Introduction

2 Anchoring

3 The Basic Deductive-Mathematical Account

4 The Genuineness Problem

5 Irrelevance

6 Relevance and Information

7 Objections and Replies
7.1 Against Relevance Logic
7.2 Too Epistemic
7.3 Informational Containment

8 Conclusion



1 Introduction

Call an explanation in which a non-mathematical fact is explained by mathematical facts: an
extra-mathematical explanation.1,2 The apparent prevalence of such explanations within
science has attracted attention in the debate over the existence of mathematical objects.3 It is
therefore surprising that few attempts have been made to develop a theory of scientific
explanation that is capable of handling run-of-the-mill scientific explanations as well as cases
of extra-mathematical explanation.

In this paper, I will not be concerned to argue for the existence of extra-mathematical
explanations. There are plenty of such defenses available.4 Instead, I will assume that there
are some such explanations and that a theory of the same is required. Importantly, I do not
view it to be a goal of developing a theory of extra-mathematical explanation that such a
theory should disabuse a sceptic about extra-mathematical explanations of her scepticism. No
matter one’s theoretical understanding of explanation there is always room for doubt, which is
just as it should be.

A theory of extra-mathematical explanation will offer an account of the explanatory relation
— R — in virtue of which mathematical facts explain physical ones. The goal of this paper is
to develop such a theory by extending the deductive-nomological theory of scientific
explanation to handle extra-mathematical cases.5 On such a theory the R-relation is a relation
of logical consequence.6 In Section Two I anchor the discussion by briefly considering some
scientific uses of mathematics. In Section Three I present a basic DN account of
extra-mathematical explanation.7 The theory is then refined in light of two problems (Sections
Four and Five). In Section Six the refined DM theory is developed in greater detail and in
Section Seven I consider some objections.

2 Anchoring

It is useful to begin by considering a clear case in which mathematics is explanatory and some
clear cases in which it is not. We should, of course, be careful about putting too much stock in
our intuitions about particular cases; intuitions should not be seen as hard boundaries for
developing a theory. Nonetheless they can provide a basis for the start of enquiry by helping to
delineate the target phenomenon. Delineating the target phenomenon with clear examples is
important, since doing so is needed to tie the phenomenon to its analysis. Once a theory of the
target phenomenon has been developed, then the relevant intuitions can be revisited, and

1This terminology is taken from Baker and Colyvan ([2011])
2Extra-mathematical explanations are typically explanations in which mathematics is

playing an explanatory role in conjunction with physical facts.
3See Baker ([2005, 2009]); Baker and Colyvan ([2011]); Bangu ([2008]); Baron (2014,

2016a, 2016b); Baron and Colyvan ([2016]); Colyvan ([2001, 2002, 2010, 2012]); Lange
([2013]); Leng ([2010]); Lyon ([2012]); Lyon and Colyvan ([2008]); Mancosu ([2001]);
Pincock ([forthcoming]); Saatsi ([2011, 2012, forthcoming]).

4See Baker ([2005], [2009], [2011]); Bangu ([2008]), Baron ([2014], [2016a], [2016b]);
Baron and Colyvan ([2016]); Colyvan ([2001], [2002], [2010], [2012]); Lange ([2013]); Lyon
([2012]); Lyon and Colyvan ([2008]).

5Baker ([2005, 2009]) has suggested such an account; so has Lange ([2013]).
6DN-style theories of scientific explanation have been resuscitated by Jansson

([forthcoming]) and Strevens ([2008]).
7See Hempel ([1965, 1948]) for the original DN account.

2



verdicts on cases where intuitions are unclear may be given.
The poster-child for extra-mathematical explanation is Baker’s ([2005]) case of the North

American magicicada. The case is by now well known and so I will be brief. Every thirteen or
seventeen years — depending on the sub-species — the magicicada arises in its adult form for
a period of around two weeks during which it eats, breeds, dies, and repeats the cycle. The
phenomenon of magicicada swarming gives rise to an important explanatory question.
Namely: why are the life-cycle lengths of the magicicada prime-numbered? The answer lies
in number theory. Because of the way primes factorize, a prime-numbered life-cycle is the
optimal strategy for avoiding predation. For instance, compare thirteen to twelve. Twelve has
factors two, three, four, six, twelve, and one. So a cicada with a twelve-year life-cycle will
overlap with predators that have periodical life-cycles of two, three, four, six, and twelve.
Thirteen, by contrast, only has factors thirteen and one, and so will overlap only with
predators that have thirteen-year life-cycles.

Further examples of extra-mathematical explanation are available.8 However, I will focus
on the cicada case for three reasons. First, as an example of extra-mathematical explanation,
that case has seen the most extensive defense to date.9 Second, the example has a good deal of
intuitive pull. The mathematics really does seem to be playing a substantive role in the
explanation. Third, the cicada case is an optimisation problem. Many of the examples of
extra-mathematical available in the literature are problems of this kind (see Baron ([2014])
and Rice ([2015])). A theory of extra-mathematical explanation focused on the cicada case
therefore promises to be widely applicable.

Compare the cicada case with the following example offered recently by Baron ([2016a], p.
451):

Suppose we want to explain why it is that a train T arrives at a station, S , at
3:00pm. The explanation is as follows: T left another station, S∗, 10 kilometres
away at 2:00pm and headed towards S at 10 kph. Obviously, this explanation
exploits some basic mathematics. Numbers are used to state the distance between
stations as well as the speed of the train and a very basic mathematical calculation
is deployed, namely: 1010 = 1.

Intuitively, the mathematics does not do any explanatory work in the explanation given.
Rather, it is physical facts about the speed of the train and the distance between stations that
fully explain why the train arrived when it did.

Steiner ([?]p. 142]steiner:am) outlines a similar case. Suppose we want to tile a rectangular
floor. It takes 500 tiles to do so. Why 500 and not 501? The answer is simple: the floor is 50
units long and 10 units wide. The number of tiles needed is equivalent to m × n where m is the

8Other examples of extra-mathematical explanation include explanations for: the shape of
honeycomb cells (Lyon and Colyvan [2008]), the structure of seeds in a flower (Lyon [2012]),
the search patterns of fully-aquatic marine predators (Baron [2014]), the minimal shape of
Plateau’s soap film (Lyon [2012]) the location of the Kirkwood gaps (Colyvan [2010]), the
Fitzgerald-Lorenz contraction of bodies at relativistic speeds (Colyvan [2002]), the instability
of high-energy galaxies (Lyon and Colyvan [2008]), the use of real-valued functions to
understand physical systems (Peressini [1997]), and the use of Hilbert spaces in quantum
mechanics as a basis for explaining quantum phenomena (Steiner [1995] and Peressini
[1997]).

9See Baker ([forthcoming, 2009]); Baker and Colyvan ([2011]); Baron ([2014]) for four
such defenses.

3



length of the floor in units and n is the width. Because 50 × 10 = 500 the number of tiles
needed is 500.

A third case has been offered by Melia:

[W]hen we come to explain [physical fact] F, our best theory may offer as an
explanation ‘F occurs because P is

√
2 metres long’. But we all recognize that,

though the number
√

2 is cited in our explanation, it is the length of P that is
responsible for F, not the fact that the length is picked out be a real number.
(Melia [2002], p. 76)

The mathematics in all three cases seems to be non-explanatory for the same basic reason:
in each case the mathematics serves as a descriptive framework within which to express the
explanation and within which to perform important calculations.10 As Peressini ([1997], pp.
218–219) argues, in cases such as these the mathematics is merely instrumental; it is not
adding anything substantive to the explanation at hand.

3 The Basic Deductive-Mathematical Account

A theory of extra-mathematical explanation will provide an account of the explanatory
relation that is present in cases like the cicada case and absent in cases like the train case. A
DN approach to scientific explanation is a good place to look for such a theory. Many current
theories of scientific explanation are tailored primarily toward modeling causal explanation.11

DN-style accounts, however, are not restricted to causal explanations. This is important as the
relation in virtue of which mathematical facts explain physical ones is not causal.12 Moreover,
as Colyvan (2010, 2012), Lyon ([2012]) and Lange ([2013]) have urged, extra-mathematical
explanations typically show that the explanandum is necessary, for some modality or other. A
theory of extra-mathematical explanation should be capable of handling the modal robustness
of these explanations. The DN theory is well-placed to do so. By making relations of
entailment carry explanations it is possible to imbue those explanations with the right kind of
modal power. A DN theory of extra-mathematical explanation also offers unique prospects for
unifying extra-mathematical explanation with intra-mathematical explanation (the explanation
of one or more mathematical facts by others). One natural thought is that explanation within
mathematics has something to do with proof on the one hand and the subsumption of
particular mathematical facts under more general mathematical patterns on the other (see, for
discussion, Lange, Lange ([2014, forthcoming])). If that’s right, then a DN theory is a good
bet.13

According to the basic Deductive-Mathematical (DM) account, extra-mathematical
explanation is a matter of subsuming explananda under mathematical facts. As in the standard
DN account, the subsumption of an explanandum under mathematical facts is modeled via the
deduction of non-mathematical claims in part from mathematical claims. The basic DM

10For further discussion of the role of mathematics in describing or ‘indexing’ physical
phenomena see Baker and Colyvan ([2011]); Saatsi ([2011, forthcoming]).

11See Salmon’s ([1984]) process theory and the counterfactual theory defended by
Woodward (2003), Woodward and Hitchock (2003), Lewis (1986) and Skow ([forthcoming]).
Some philosophers suggest ways of extending their accounts to handle non-causal
explanations in science. See e.g., Woodward ([2003]) and Salmon ([1984]).

12Lyon (2012) makes this point.
13For discussion of intra-mathematical explanation see Colyvan et al. ([2017]).

4



account can be stated as follows:14

The Basic DM Theory of Extra-Mathematical Explanation

1 Extra-mathematical explanations are sound arguments.

2 The conclusion of an extra-mathematical explanation is a proposition stating the
physical phenomenon to be explained.

3 Among the premises of an extra-mathematical explanation there must be at least one
mathematical claim.

4 If the mathematical claim were removed from the premises of an explanatory argument,
then the argument would become invalid.

It is useful to apply the theory to an actual case. It is straightforward to formulate Baker’s
cicada case as a deductive argument (see Baker ([2005], p. 233)):

[P1] Having a life-cycle period which minimises intersection with other (nearby / lower)
periods is evolutionarily advantageous.
[P2] Prime periods minimise intersection (compared to non-prime periods).
[P3] If having a life-cycle period which minimises intersection with other (nearby / lower)
periods is evolutionarily advantageous, and prime periods minimise intersection (compared to
non-prime periods) then organisms with periodic life-cycles are likely to evolve periods that
are prime.
[P4] Cicadas in ecosystem-type, E, are limited by biological constraints to periods from
fourteen to eighteen years.
[P5] If organisms with periodic life-cycles are likely to evolve periods that are prime and
cicadas in ecosystem-type, E, are limited by biological constraints to periods from fourteen to
eighteen years, then cicadas in ecosystem-type, E, are likely to evolve seventeen-year periods.

[P6] Cicadas in ecosystem-type, E, are likely to evolve seventeen-year periods.

The argument satisfies the tenets of the basic DM theory and as such is classified as a genuine
case of extra-mathematical explanation, which is the right result. In what follows I will refine
the basic DM account by addressing a series of difficulties for the view. The result will be a
more robust theory that may be used as a basis for future work.

14A note on terminology: a ‘mathematical claim’ is a true or false proposition about a
mathematical fact, such as the claim that there are an infinite number of primes. Mathematical
claims are represented by sentences that use mathematical language. The deduction of a
non-mathematical claim from mathematical claims will involve a group of premises that are
constituted by mathematical sentences expressing those claims and a conclusion that is
constituted by a non-mathematical sentence (a sentence using no mathematical language) that
expresses a non-mathematical claim (such as a proposition about empirical reality).

5



4 The Genuineness Problem

Here’s the first problem. It is a straightforward matter to formulate cases like the train case
outlined in Section Two as deductive arguments that satisfy the core tenets of the basic DM
account:

[P1] T left S∗ at 2:00 pm.
[P2] S∗ is 10 kilometres away from S .
[P3] T is travelling at 10 kph.
[P4] For any number m mm = 1.
[P5] If for any number m mm = 1, then

10
10 = 1.

[P6] If T left S∗ at 2:00 pm, S∗ is 10 kilometres away from S , T is travelling at 10 kph, and
10
10 = 1, then T arrives at S at 3:00pm.

[P7] T arrives at S at 3:00pm.

By formulating the train case in this fashion the point is not that the DM theory incorrectly
classifies the case as an extra-mathematical explanation, thereby violating our intuitions
(though it certainly does that). The point, rather, is that the ease with which such an argument
may be formulated is telling. A little reflection reveals that most (if not all) uses of
mathematics in science can be formulated into an argument in much the same manner. So the
basic DM theory lacks the resources to uphold a meaningful distinction between explanatory
and non-explanatory uses of mathematics in science. Call this: the genuineness problem.

In order to solve the genuineness problem, a further constraint on the basic DM account is
needed. Here’s my suggestion: claims concerning the non-mathematical facts that one is
attempting to explain must be essentially deducible from the mathematical claims at issue.
This ‘essential deducibility’ constraint may be formulated as follows (note that a ‘physical
sentence’ is a sentence that expresses a claim about physical reality and that uses no
mathematical language):

Essential Deducibility Constraint [EDC]: A non-mathematical claim P is
essentially deducible from a premise set S that includes at least one mathematical
sentence M just when there is a sound derivation of P from S and there is no
sound derivation of P from a premise set S∗ that includes only physical sentences.

When a non-mathematical claim is essentially deducible from a set of premises that includes
mathematical claims, the argument is a genuine extra-mathematical explanation. When the
non-mathematical claim is not essentially deducible in this manner, because there is an
alternative derivation that includes only physical claims, the derivation of the
non-mathematical claim in part from mathematical claims is not a genuine extra-mathematical
explanation.

Some fine-tuning of the constraint is called for. What it is for mathematics to be playing a
genuinely explanatory role in science can’t simply be a matter of there being no alternative
way of getting to the physical facts. Rather, presumably even if an alternative route is
available, it may sometimes be the case that the way via mathematics is superior to the
non-mathematical way. This idea may be captured by sharpening the essential deducibility
constraint as follows:

Sharp Essential Deducibility Constraint [SEDC]: A non-mathematical claim P is
essentially deducible from a premise set S that includes at least one mathematical

6



sentence M just when there is a sound derivation of P from S and either there is
no sound derivation of P from a premise set S∗ that includes only physical
sentences or all sound derivations of P from premise sets S 1...S n each of which
includes only physical sentences are worse than the mathematical derivation.

The SEDC presumes that derivations can be ranked in terms of overall quality. For the
constraint to yield the desired result derivations involving mathematics must dominate at least
some such orderings. To give the SEDC content something must be said about what the
relevant ranking is. We need criteria by which derivations may be ranked; criteria that reflect
the good-making features of the physical or scientific explanations that the derivations encode.
Exactly what these criteria are is therefore beholden to our best account of the good-making
features of physical or scientific explanations. What those good-making features are is
controversial. For now it will be sufficient to show how one might rank derivations with
respect to two of the most widely accepted good-making features of such explanations:
simplicity and unity.

Here’s the idea. A sound argument A for conclusion C is simpler than a sound argument A∗
toward the same conclusion when A has fewer distinct premises than A∗. A sound argument A
for conclusion C has a greater degree of unity than a sound argument A∗ toward the same
conclusion when A’s premises may be used to establish a greater range of conclusions other
than C than can A∗’s premises. The best derivations are those that strike the optimal balance
between simplicity and unity understood in this sense. The arguments that strike this balance
represent those explanations that are the simplest and that have the most unificatory power,
which will be among the best explanations for the phenomenon of interest. Note that the move
here is by no means new: the trade-off is similar to the one found in Lewis’s ([1994])
best-systems account of law-hood. Unity as I use it here is Lewis’s notion of strength.

Both simplicity and strength (unity) are contested notions. So I need to say a bit more.
First: simplicity. Simplicity has been the subject of a great deal of criticism in the philosophy
of science. In the wake of this critique, an important strategy for justifying simplicity has
emerged. This strategy is broadly probabilistic in nature: simpler theories are more likely to
be true.

One version of the probabilistic strategy — the likelihood account — may be sketched as
follows. First, given Bayes’ theorem, the probability of a theory T (simple or complex) given
evidence E is:

P(T |E) =
P(T ) × P(E|T )

P(E)
(1)

Next, note that simpler theories are compatible with fewer items of evidence. This is because
simpler theories tend to have fewer adjustable parameters. With fewer adjustable parameters,
there is less scope for a theory to be modified so that it fits the data (see Huemer ([2009], pp.
221–222) and Forster and Sober ([1994], pp. 10–11). Suppose, then, that we have a simple
theory S and a complex theory C. Suppose, further that S is compatible with and neutral
between two items of evidence E1 and E2, whereas C is compatible with and neutral between
four items of evidence E1, E2, E3, and E4. Now, consider a particular piece of evidence, E1.
Suppose that E1 and only E1 is observed. Which theory is more likely to be true? Well,
P(E1|S ) =

1
2 , whereas P(E1|C) =

1
4 . Assuming the prior probabilities of S and C are equal, it

follows by [A] that P(S ) > P(C).15

15This example is taken from Huemer ([2009], pp. 221–222).

7



This is not the only probabilistic justification for why it is that simpler theories tend to be
more likely. A range of other strategies have been offered and developed in detail by Sober
([2016]).16 I won’t survey these strategies here. All that matters is that the SEDC uses a type
of simplicity that is ultimately open to probabilistic justification, since then it is in line with
the use of simplicity in science more generally. And indeed it does: the type of simplicity used
by the SEDC concerns the number of distinct premises within a deduction. Translated into the
explanatory key of the DM theory, the number of premises corresponds to the number of
distinct claims that constitute an explanation. Essentially, an explanation is simpler when it
makes fewer claims. An explanation that makes fewer claims, however, will be compatible
with fewer items of evidence. By making fewer claims, it has fewer adjustable parameters and
so will be harder to fit to the data. By the same Bayesian reasoning, then, explanations that are
simpler in the sense just described are more likely to be true. So the likelihood account
applies. But so too will other probabilistic justifications, since those justifications typically
apply to the type of simplicity under discussion.

This brings us to strength. There is a well-known problem with Lewis’s notion of strength.
The problem is that the strength of a deduction is beholden to the system of basic predicates
that is used to underwrite the deduction Cohen and Callender ([2009], p. 6). If the strength of
a deduction is beholden to a choice of expressive resources, then it will be difficult to compare
DM arguments in terms of their strength without already settling on a choice of basic
predicates. But how do we do that? Lewis ([1983]) maintained that nature provides us with a
privileged class of predicates. The predicates in question are those that correspond to the
perfectly natural properties.

There are two problems with Lewis’s solution. First, the perfectly natural properties are, for
Lewis at least, the fundamental properties. This makes it very difficult to understand strength
in the context of sciences that deal in the non-fundamental (Schrenk [2014], pp. 1791–1792).
Second, it is unclear how to extend the concept of a perfectly natural property to mathematics.
One option is to appeal to fundamental properties again. But extra-mathematical explanations
often deal with non-fundamental mathematical properties (such as certain number theoretic
properties).

Cohen and Callender ([2009]) offer an alternative to Lewis’s solution. Let strength be
relativized to a choice of basic predicates. Let the choice of basic predicates be set by the
science within which the explanation is formulated. So, for example, within biology, let the
basic predicates be biological ones; within physics, let the basic predicates be the predicates of
fundamental physics and so on. Having relativized strength in this manner, conduct the
trade-off between strength and simplicity with respect to that choice. Finally, accept that there
are many strongest systems of derivation, each relativized to a scientific domain.

Cohen and Callender’s solution can be extended for use by the DM theory. Let us allow that
basic mathematical predicates form part of the expressive stock for determining strength. With
respect to derivations involving mathematics, we then let the choice of basic predicates be
partially set by the area of mathematics being used and partially set by the domain of science.
So, for instance, if one is working in biology using number theory, then let the basic predicates
be those found in biology and number theory; if one is working in physics and graph theory,
then let the basic predicates be physical and graph-theoretic and so on. The strength of
derivations can then be compared with respect to a choice of basic predicates, with the
strongest being the best for a mixed scientific/mathematical domain.

How much of a mathematical theory is required in order to delimit the mathematical

16See also Jansson and Tallant ([2016]).

8



vocabulary at issue? It may be that the full pure mathematical theory is invoked. For example,
Steiner ([1995], p. 149) and Peressini ([1997], p. 221) both argue that for some mathematical
explanations of quantum phenomena, the entire Hilbert space formalism that underwrites the
explanation will be needed.17 If that’s right and, moreover, the full pure mathematical theory
is generally required, then the deductions at issue may invoke all of the pure mathematical
inferences that correspond to the relevant pure theory. A case in which a physical claim is
deduced from mathematics may therefore be very strong indeed. But this kind of strength is
not of the right type for comparing mathematical explanations of physical phenomena. We
want to compare these explanations in terms of the consequences they have for physical
claims only. Accordingly, a further modification to the Cohen and Callender solution is
required. The notion of strength at issue must be restricted to the deduction of
non-mathematical claims. This prevents the purely mathematical implications from
introducing unwanted noise into our evaluation of the relative strength of DM arguments.

While adopting Cohen and Callender’s view helps, doing so gives rise to another difficulty.
Suppose we want to explain a particular phenomenon P. There are two choices of basic
predicates available for doing so. One choice uses purely physical language. The other uses a
mixture of mathematical and physical predicates. Now, suppose that a derivation of P is
possible with respect to both choices. In the purely physical case, the strongest derivation of P
uses only physical predicates; in the mixed mathematico-physical case, the strongest
derivation uses mathematical and physical predicates. In this situation, the mathematics might
not be playing an explanatory role; for there is an explanation available for P that makes no
use of mathematics. However, because the two derivations are conducted using different
expressive resources, the strength of the two deductions cannot be compared. Because the
strength of the two derivations cannot be compared, we cannot say that one explanation is
better than the other, and so we cannot determine whether the mathematical derivation is
better than the non-mathematical one. So we don’t know if the mathematics is playing a
genuine explanatory role. The SEDC doesn’t apply in this case. But this is exactly where it
should apply.

To address this problem I propose a further modification to the SEDC:

Razor-Sharp Essential Deducibility Constraint [REDC]: A non-mathematical
claim P is essentially deducible from a premise set S that includes at least one
mathematical sentence M just when for an appropriate choice of expressive
resources there is a sound derivation of P from S and either for the same choice of
expressive resources there is no sound derivation of P from a premise set S∗ that
includes only physical sentence or all sound derivations of P from premise sets
S 1...S n each of which includes only physical sentences are worse than the
mathematical derivation or for all appropriate choices of expressive resources the
best derivations use M.

We supervaluate over appropriate choices of expressive resources; a choice is ‘appropriate’
when it is licensed by the mathematical or scientific domains within which the explanation is
formulated. When M dominates no matter which choice of basic predicates is used, then M is
genuinely explanatory. If, however, for a choice of expressive resources, there is a way of
deriving P that makes no use of M and that bests any derivation that uses M, then M is not
explanatory.

17The more general issue of how much of a pure theory is needed in a given case of
explanation is discussed in Peressini ([2010]) and Steiner ([1992]).

9



While the REDC provides a more nuanced account of strength and simplicity, it faces a
further difficulty. Consider a DM argument in which P is derived, in part, from a mathematical
claim M. Call the argument α. Suppose α satisfies the REDC. Suppose, also, that this is
sufficient for α to count as a genuine extra-mathematical explanation. In virtue of satisfying
the REDC, it also follows either that there is no alternative derivation (and thus explanation
of) P that does not feature M or that all alternative derivations that do not feature M are
inferior (either for a choice of expressive resoures, or across all choices). But M is
indispensable to an explanation when these same conditions are met. So, by REDC, any case
of indispensability for M is also a case of genuine extra-mathematical explanation. This leaves
no room for M to be indispensable to an explanation, and yet for M to play a merely
descriptive role in that explanation. In short, the REDC threatens to collapse the distinction
between descriptive and explanatory indispensability (and thus between explanation and
indispensability more generally).

To avoid this outcome, a further constraint must be imposed on the basic DM theory. In
order to state the constraint, however, I require the notion of informational containment
developed in Sections Four and Five, which is used to address the second problem facing the
basic DM theory. Let us therefore set aside the genuineness problem for a moment and return
to it with a concept of informational containment in hand.

5 Irrelevance

The second problem facing the basic DM theory focuses on relevance. Consider the following
argument:

[P1*] Having a life-cycle period which minimises intersection with other (nearby / lower)
periods is evolutionarily advantageous.
[P2*] 2 + 2 = 4.
[P3*] If having a life-cycle period which minimises intersection with other (nearby / lower)
periods is evolutionarily advantageous, and 2 + 2 = 4 then organisms with periodic life-cycles
are likely to evolve periods that are prime.
[P4*] Cicadas in ecosystem-type, E, are limited by biological constraints to periods from
fourteen to eighteen years.
[P5*] If organisms with periodic life-cycles are likely to evolve periods that are prime and
cicadas in ecosystem-type, E, are limited by biological constraints to periods from fourteen to
eighteen years, then cicadas in ecosystem-type,E, are likely to evolve seventeen-year periods.

[P6*] Cicadas in ecosystem-type, E, are likely to evolve seventeen-year periods.

All I have done is swap one mathematical claim for another. Since the mathematical claim
expressed in [P2*] is true, [P2*] and [P3*] are both true, and since the logical form of the
argument remains intact, the argument is valid (and so sound if the original argument is). The
argument also satisfies the core tenets of the basic DM account, since the explanandum is
derived, in part, from mathematics. Finally, the argument satisfies the REDC just when
Baker’s original argument does. So it seems we are forced to treat the above argument as one
that represents a genuine extra-mathematical explanation. And yet that is the wrong result: the
mathematical claim at [P2*] — that 2 + 2 = 4 — is irrelevant to cicada life-cycle length.

Call this the irrelevance problem. Note that, as with the genuineness problem, the worry is

10



not that the DM theory contradicts our intuitions about what is relevant and what is not for a
given explanation (though it certainly does that). The problem is that any and all mathematical
claims can be added to the cicada case (or, indeed, any DM argument) in the same manner.
But it is not the case that all mathematical claims are relevant to every explanandum. So
something has gone wrong: the DM theory lacks the capacity to sort relevant mathematical
claims from irrelevant ones when it comes to explanation.

The difficulty is quite similar to a familiar problem facing the standard DN account of
scientific explanation. For instance, consider the following argument outlined by Salmon
(1971, p. 34):

[P1] All males who take birth control pills regularly fail to get pregnant.
[P2] John Jones is a male who has been taking birth control pills regularly.

[P3] John Jones fails to get pregnant.

Clearly, the above argument does not represent a genuine scientific explanation. John Jones’s
failure to get pregnant is not explained by his birth-control regimen. Standard DN accounts
must be modified in order to handle this kind of case. The problem facing the basic DM
account is that the standard strategies available for addressing the John Jones case are not
available for ruling out arguments like the specious cicada argument and so cannot be used to
differentiate relevant from irrelevant mathematical claims.

There are, broadly speaking, two such strategies. The first is to apply a further, causal
condition of adequacy over DN arguments. So, for instance, in the John Jones case, it is not
true that taking birth control pills causes John Jones’s failure to get pregnant. It is, rather, the
fact that he lacks the physiological traits needed to get pregnant that causes his failure to get
pregnant. Accordingly, it is possible to rule that the John Jones argument is not a genuine
scientific explanation because only those arguments that correspond to the causal facts of the
explanatory situation are genuine scientific explanations.

The second strategy is to provide a more precise account of what it is to be a law of nature.
The John Jones case relies on one’s account of what it is to be a law of nature being
sufficiently broad to include the generalisation that all males who take birth control pills
regularly fail to get pregnant. By supplementing the standard DN account with a more
stringent account of what it is to be a law, one may be able to rule that the first premise of the
John Jones case does not, in fact, correspond to a law. This would then disqualify the
argument from being an explanation.

Evidently, the causal strategy is of no use in patching up the basic DM theory. There is no
difference in the causal facts being tracked by Baker’s cicada argument as compared to an
argument like the specious cicada argument outlined above. In so far as either argument tracks
the causal facts of the situation, they both do. There is no further causal relationship between,
say, prime numbers and cicada life-cycle length that Baker’s cicada argument captures but that
the specious argument misses. Indeed, a moment’s reflection reveals that no causal constraint
whatsoever can be used to solve the irrelevance problem for the basic DM theory:
mathematical facts do not cause physical facts and so there is no causal difference between
two DM arguments that differ only with respect to the mathematical claims they deploy.

Providing a more precise account of what it is to be a law of nature won’t obviously help
either, since the difficulty posed by arguments like the specious cicada argument has nothing
to do with the laws. It is not because ‘2 + 2 = 4’ fails to be a law, whereas ‘prime periods
minimise intersection (compared to non-prime periods)’ is a law that we can rule against

11



arguments like the specious cicada argument. Neither of these claims are laws of nature in the
relevant sense, they both correspond to mathematical facts. So there is no way to sort relevant
from irrelevant mathematics by tightening the notion of a law.

Now, one might seek to broaden the concept of a physical law of nature to include
statements that feature mathematical claims. Indeed, Baker ([2005]) makes a suggestion along
these lines. So, for instance, compare [P3] with [P3*] (repeated below):

[P3] If having a life-cycle period which minimises intersection with other (nearby
/ lower) periods is evolutionarily advantageous, and prime periods minimise
intersection (compared to non-prime periods) then organisms with periodic
life-cycles are likely to evolve periods that are prime.

[P3*] If having a life-cycle period which minimises intersection with other
(nearby / lower) periods is evolutionarily advantageous, and 2 + 2 = 4 then
organisms with periodic life-cycles are likely to evolve periods that are prime.

It could be argued that while [P3] is a candidate to be a physical law of nature, [P3*] is not.
In order to turn this move into a solution to the problem at hand, however, one must strengthen
the DM theory with the following requirement:

(3*) Among the premises of an explanation there must be at least one
mathematical claim and it must feature in a law of nature.

By strengthening the DM theory with (3*), it may be possible to rule out arguments like the
specious cicada argument: that argument is not an extra-mathematical explanation because
[P3*] is not a law of nature, and ‘2 + 2 = 4’ does not feature in any other laws of nature within
that argument. There are, however, two difficulties with this suggestion. First, as Baron
([2016a], pp. 466–467) has argued, the same problem can be formulated for the laws
themselves. We can always take a putative law like [P3] and replace the mathematics in [P3]
with something irrelevant to produce a claim that is just as good a candidate to be a law of
nature. We just do not have an account of physical laws available that will rule out [P3*] while
ruling [P3] in. Second, (3*) may be too strong a requirement to place on the DM theory. As
Lange ([?]p. 505]lange:wmsedm) has argued, one of the distinctive features of
extra-mathematical explanations is that they have the capacity to explain physical phenomena
without involving any laws of nature. Such explanations can therefore show an explanandum
to hold even if the laws were different. If the mathematics in an extra-mathematical
explanation must always be routed through the laws, however, then this distinctive feature of
extra-mathematical explanation will be lost. At best, such an explanation can show an
explanandum to hold of physical necessity.

It seems, then, that standard solutions to irrelevance worries for traditional DN accounts of
explanation won’t generalize to the DM account. So a new solution is required. Here’s my
(somewhat radical) proposal: to solve the problem, modify the logical foundations of the DM
theory. Instead of formulating the DM theory in the same logical terms as the DN theory —
namely, using classical logic — we may instead formulate the DM theory using a relevance
logic. In particular, the relevance logic R.18,19 Relevance logic places a relevance constraint on

18R is needed because R and its fellows make use of the containment relation ‘v’, which is
needed to handle the notion of informational containment described in Section Six.

19The use of relevance logic as a basis for understanding mathematical explanation has
already been suggested by Colyvan et al. ([2017]) in the context of intra-mathematical
explanation.

12



both the semantic relation of consequence that connects premises to conclusions and the
conditional ‘→’. Within relevance logic, Γ |= ∆ only if Γ is relevant to ∆ and Γ → ∆ only if Γ
is relevant to ∆.20 Exactly what it means to say that Γ is relevant to ∆ will be considered in
greater detail in the next section. For now, it is enough to see how shifting toward a relevance
logic helps with the irrelevance problem.

Consider, again, the specious cicada argument outlined above. This argument seems
specious precisely because ‘2 + 2 = 4’ does not appear to be relevant to the life-cycle length of
the periodical cicadas. A relevance logic has the potential to satisfy this basic idea. As we
have seen, if [P3] in Baker’s formulation of the cicada argument is swapped for [P3*], and
[P2] for [P2*] a sound argument is produced (so long as the argument with [P3] and [P2] was
already sound). But while classical logic is blind to the difference between [P3] and [P3*]; a
relevance logic is not. A relevance logic vindicates [P3] but not [P3*]: [P3] is true because
each part of the antecedent is relevant to the consequent; [P3*] is false because some part of
the antecedent is irrelevant to the consequent. Note that to deliver this result, what we must
demand is that the antecedent be wholly relevant to the conclusion, in this sense: no part of the
antecedent is irrelevant to the consequent. The same demand may be imposed on the
consequence relation from premise to conclusion, as follows: no part of the set of premises is
irrelevant to the conclusion. Consequence under this demand is no-longer simply a matter of
necessary truth-preservation from premise to conclusion. Rather, consequence is now a matter
of relevantly constrained necessary truth-preservation: both truth and relevance must be
preserved from premise to conclusion.

Call an argument that obeys the strictures of a relevance logic, an R-Argument, then
according to the refined DM theory of extra-mathematical explanation, extra-mathematical
explanations are sound R-Arguments. The refined DM theory of extra-mathematical
explanation has an added advantage. Explanations are non-monotonic. If A explains B, then it
is not the case that A and C explain B, for any C whatsoever. Classical consequence, however,
is monotonic. So one can produce all kinds of junk arguments that satisfy the strictures of the
basic DM account simply by exploiting the monotonicity of the classical consequence
relation. The consequence relation of relevance logic is non-monotonic. So the logical
foundations of the refined DM theory accord more closely with the logical features of
explanation more generally. It is a straightforward matter to demonstrate the
non-monotonicity of relevance logic, but I will spare the reader the technical details.21 Instead,
we can understand in an informal sense why a relevance logic is non-monotonic as follows.
Suppose, as I have suggested, that the premises need to be wholly relevant to the conclusion:
no part of the premise set is irrelevant to the conclusion. Well, it follows that one cannot add
premises to the premise set willy-nilly and expect validity to be preserved. For if one adds an
irrelevant premise, then the validity of the argument will be undermined, since the premises
will no-longer be wholly relevant to the conclusion. So the logic is non-monotonic.22

20See Andersen and Belnap ([1975]); Goldblatt and Mares ([2006]); Mares ([2009]); Mares
([2004]); Priest ([2002]) and Restall ([1996]) for the formal details.

21Relevance logics are non-monotonic because they jettison weakening. See Restall and
Dunn ([2002]).

22Using relevance logic because it is non-monotonic mirrors a solution offered by Waters
([1987]) to the irrelevant conjunction problem for hypothetico-deductivism, which may be
stated as follows (see Skyrms ([1992]) for an overview of the problem). For background
theories T , for hypothesis H and evidence e if e confirms H then: (i) T and H are consistent,
(ii) T, H ` e, and (iii) T 0 e. When T , H and e satisfy these three conditions, for any A
whatsoever, T , (H&A) and e also satisfy the conditions. Waters proposes to address the

13



As it stands, the refined DM theory of extra-mathematical explanation may not seem
particularly plausible. Consider the concept of relevance that is at work in the statement of the
refined DM theory. What is it for the antecedent of a conditional to be relevant to its
consequent? Similarly, what is it for the premises of an argument to be relevant to their
conclusion? Here’s one answer: Γ |= ∆ only if Γ is explanatorily relevant to ∆ and Γ → ∆ only
if Γ is explanatorily relevant to ∆. If this answer is correct, then the refined DM theory of
extra-mathematical explanation is no good. Recall that our goal in producing a theory of
extra-mathematical explanation was to provide some account of the relation, R, in virtue of
which mathematical facts explain non-mathematical facts. But if the notion of relevance that
is operative in a relevance logic just is explanatory relevance, then it would seem that an
account of R is the very thing we need to interpret the concept of relevance that the logic
deploys. In short, if ‘relevance’ is understood to be ‘explanatory relevance’ then the refined
DM theory is painfully circular.

In the following section I will outline one, recent interpretation of the relevance relation that
does not treat relevance as explanatory relevance. In Section Seven I will then raise and
address two objections for that interpretation when it is used as a basis for clarifying the
refined DM theory of extra-mathematical explanation. First, however, it is worth pausing to
consider one option for understanding relevance that I won’t pursue. This is to understand
relevance syntactically. A syntactic understanding of relevance treats relevance entirely in
terms of variable sharing. Very roughly, on such a view, what it is for the premises of an
argument to be relevant to the conclusion of that argument is for there to be (at least one)
propositional variable that appears in both the premises of the argument and the conclusion of
that argument. A variable sharing understanding of relevance for DM arguments might
therefore be provided as follows: it is a necessary condition over DM arguments that at least
some of the mathematics appearing in the premises of those arguments also appears in the
conclusion. This way of interpreting relevance will successfully rule out arguments like the
spurious cicada argument outlined above, but it ultimately won’t do. Many cases of
extra-mathematical explanation don’t feature any mathematics in the conclusion.23 So variable
sharing of the kind described for mathematical claims may not be assured. Indeed, as Bangu
([2008]) has argued, if mathematical facts do appear in the explanandum of a putative
extra-mathematical explanation, then that explanation is left open to the charge that it is
actually an intra-mathematical explanation: an explanation of one mathematical fact by
another.24 It follows that in order for the DM theory to correctly handle extra-mathematical
explanations as such, an alternative account of relevance is needed.

6 Relevance and Information

In providing an interpretation of the concept of relevance that appears in relevance logic the
aim is not to provide a semantics for relevance logic. The standard semantics for relevance

irrelevant conjunction problem by using relevance logic, rather than classical logic, as the
basis for hypothetico-deductivism.

23Colyvan’s (2010) example of the Kirkwood gaps is like this.
24Baker ([2009]) recognizes this difficulty and, in recent work, has reformulated the cicada

case without any mathematical facts in the explanandum.

14



implication is the Routley-Meyer ternary semantics.25,26 Rather, by ‘interpretation’ I mean a
philosophical account of what it means to say that the premises of an argument must be
relevant to its conclusion, and the antecedent of a conditional must be relevant to its
consequent. There is no one interpretation of relevance that all logicians accept. Nevertheless,
it is possible to discern a common thread that runs through extant interpretations; an element
that constitutes something of a consensus about how to understand relevance implication. This
is the idea that relevance is to be understood in terms of information. The kind of information
at issue is semantic information — information about something — and has its roots in the
information theory coming out of the work of Dretkse ([1981]), Israel and Perry ([1990]), and
Barwise and Seligman ([1997]).

There are various ways to interpret relevance logic through the lens of information theory. It
is not necessary to lay out the full details of each informational account of relevance logic
here. For now, I will proceed with a rough and rather broad account of relevance logic in
terms of informational containment. What it means to say both that Γ |= ∆ only if Γ is relevant
to ∆ and that Γ → ∆ only if Γ is relevant to ∆ is this: (i) all of the information contained within
∆ is contained within Γ and (ii) each member of Γ contains some part of the information in ∆.
A proposition gets its informational content from the part of the world that makes it true. So,
for instance, the proposition that the window is smashed by the rock contains information
about window smashings and rocks, and gains that information by being made true by a
particular window-smashing event. Similarly, mathematical claims such as 2 + 2 = 4 get their
information from the mathematical facts about two and four that make that claim true.

What is semantic information metaphysically speaking? There isn’t much consensus on this
either. However, despite lacking a fully worked-out metaphysics for semantic information, the
notion receives rather heavy use in relevance logic and information theory nonetheless. I hope
I won’t be too disparaged as I join the multitudes. Besides, semantic information is clearly
something that we all have some grasp on. Semantic information is what we’re talking about
when we say things like: ‘my map provides information about the whereabouts of local bars’
or ‘this novel contains information about the history of World War II’ or ‘your description of
the solar system gives me a lot of astronomical information to work with’ or even ‘what you
just said was very informative’. It’s that kind of information that is being used to interpret the
concept of relevance, and it is that kind of information that is ultimately underwriting the
refined DM theory of extra-mathematical explanation. Note also that the concept of
explanatory information already has currency in the literature on scientific explanation.27

Explanatory information is just a particular kind of information, namely the kind of
information that features in explanations. What the refined DM theory does is provide a
partial account of what kind of information explanatory information is: it is information that
features in relationships of informational containment where those relationships involve

25See Routley and Meyer ([1973]); Routley and Meyer ([1972a]) and Routley and Meyer
([1972b]). For discussion see Beall et al. ([2012]).

26The truth-conditions for the relevant ‘→’ are to be analysed in terms of a ternary
accessibility relation Rxyz over worlds u, v, w (the semantics extends in the usual way to
Γ |= ∆):

w |= φ → ψ if and only iff for each u, v where Ruwv, if u |= φ then v |= ψ. (Restall
1996a, p. 464)

27E.g., the ‘explanatory store’ in Kitcher ([1989]) and the ‘ideal explanatory text’ in Railton
([1978, 1981]).

15



mathematical facts essentially.
What is informational containment? As before, no agreed-upon theory is available but,

again, a basic grip on this notion can be easily gained. Informational containment is the
phenomenon we have in mind when we say things like ‘my map and your map contain the
same information’ or ‘the information provided by the defendent does not contain the same
information as that provided by the prosecution’ or ‘the second edition contains all of the
information about World War II that the first edition contained, plus some extra information
about Russia’s involvement’. Informational containment is a relation between pieces of
information, which we may call: information states. An information state i contains the
information in a distinct information state j when information state j is a part of information
state i.

There is a great deal more to say about semantic information and about informational
containment, but I won’t attempt to say it here. All that matters is that both semantic
information and the relation of informational containment are phenomena that are fairly
familiar to us and to which we often appeal. So despite lacking a theory of these phenomena, I
see no problem with making use of informational containment in the present context. Indeed,
issues to do with the nature of semantic information and of informational containment are best
taken up by information theorists; whatever the correct account of these phenomena turns out
to be, I will take that account and use it to intepret relevance logic and, by extension, the
refined DM theory of extra-mathematical explanation. Importantly, the relation of
informational containment is not analysed in terms of explanation. Thus, because the refined
DM theory analyses extra-mathematical explanation in terms of informational containment
(plus some further features) the threat of circularity is avoided.

Once we interpret relevance using a relation of informational containment, the power of the
refined DM theory to solve the irrelevance problem raised in the previous section becomes
evident. That problem arises because we can add any and all mathematical claims into a sound
argument that seems to be a genuine extra-mathematical explanation and produce a second,
sound argument that is then classed as an extra-mathematical explanation as well; the basic
DM theory is blind to the distinction between relevant and irrelevant mathematics. We now
have a diagnosis of why spurious arguments produced in this manner are spurious. Consider
the cicada case. Suppose, as discussed previously, we swap facts about prime numbers out for
the fact that 2 + 2 = 4. 2 + 2 = 4, as we can clearly see, is irrelevant to the fact that we are
attempting to explain. What this means is that 2 + 2 = 4 does not contain the right
information. That seems exactly right for the case at hand: [P3*] in the specious cicada
argument is false precisely because not all the information in the consequent (about the prime
numbered life-cycles) is contained in the antecedent of the conditional, since it makes no
mention of prime numbers at all.

The following variation on the specious cicada argument can also be handled. Consider
[P3**]:

[P3**] If having a life-cycle period which minimises intersection with other
(nearby / lower) periods is evolutionarily advantageous, and prime periods
minimise intersection (compared to non-prime periods) and pythagoras’ theorem
is true then organisms with periodic life-cycles are likely to evolve periods that
are prime.

[P3**], like [P3*], can be used to formulate a specious version of the cicada argument (this is
the type of case alluded to in Section Five in which the monotonicity of classical consequence
is exploited to produce a junk argument). According to the refined DM theory [P3**] is false

16



because not only the information in the consequent is contained in the antecedent of the
conditional, since Pythagoras’ theorem appears in the antecedent and the consequent makes
no mention of geometry at all.

In general, then, the refined DM theory uses the difference between relevant and irrelevant
mathematical information as a basis for differentiating between relevant and irrelevant
mathematics in any given explanation. The notion of informational containment can also be
used to provide a full solution to the genuineness problem raised in Section Three. As we saw,
using the REDC to manage the genuineness problem threatens to collapse the distinction
between the explanatory and descriptive indispensability of mathematics. But now consider,
again, an argument α in which a mathematical claim M is used to derive a non-mathematical
claim P. Suppose that M in α contains information I that is also contained within the
explanandum P and it is in virtue of carrying this information that P is derivable from M.
Finally, suppose that some part of the information I is descriptive information about a physical
system. Call this descriptive information: i ⊇ I (see Section Six for an account of this
descriptive information and how it differs from non-descriptive information).

Now, perform the following ‘informational test’. Imagine removing i from the total
informational content of M whilst holding the rest of the informational content of M fixed.
Would M still contain some of the information that is contained within the explanandum P? If
the information in i exhausts the information that M carries regarding P, then M fails the
informational test and is only contributing descriptive information to the explanation at hand.
In this situation, M is playing a merely descriptive role in α. If, however, the information in i
does not exhaust I, then M passes the informational test and carries non-descriptive
information regarding the explanandum, information that is used to derive the explanandum
and thus explain it. In this situation M plays a non-descriptive role in the explanation.

Using the informational test, conditions for when a DM argument is a genuine
extra-mathematical explanation can be given as follows:

[Genuineness] A DM argument is a genuine extra-mathematical explanation when
it satisfies the REDC and at least one mathematical claim in that argument passes
the informational test.

The genuineness condition enables the differentiation of explanatory and descriptive
indispensability. To see this, consider two cases.

First, consider again α. Suppose that there is no alternative way of deriving P and that α
satisfies the conditions laid down by the basic DM theory as well as the REDC. Given that
there are no alternative derivations of P, M is indispensable to explaining P. But now suppose
that M fails the informational test. Then, despite the fact that M is indispensable to explaining
P, it is only contributing descriptive information. Suppose, however, that M passes the
informational test. Then, M is both indispensable for explaining P and is playing a genuine
explanatory role in the explanation.

Second, suppose that α is not the only way of deriving P and, moreover, that there is an
alternative way of deriving P from non-mathematical facts alone, it is just that the
mathematical derivation is superior to the non-mathematical derivation. Suppose also that α
satisfies the conditions laid down by the basic DM theory as well as the REDC. If M fails the
informational test, then M is both indispensable and merely descriptive; if M passes the
informational test then M is indispensable and genuinely explanatory. As with the previous
case, a robust distinction between explanatory and descriptive indispensability is upheld.

17



7 Objections and Replies

That concludes my presentation of the DM theory. I will now consider and rebut three
objections against the view.

7.1 Against Relevance Logic

Here’s the first objection: any deviation from classical logic should be treated with
suspicion.28 Such suspicion thus falls upon the refined DM theory as well. There are two
things to say here. The first is flat-footed: while classical logic has its place, maybe it just isn’t
the kind of thing that we should be using to model explanation. This doesn’t mean that
classical logic needs to be abandoned entirely: classical logic can still legitimately be viewed
as the correct account of the consequence relation. Rather, the point is just that classical logic
does not provide an appropriate framework for modeling the explanatory relation; relevance
logic is needed instead. The second response is more concessive: the refined DM theory can
be reformulated entirely within classical logic, should one wish to do so. Here’s how: take the
space of arguments that are classically valid. Now take the sound arguments from that space
that satisfy the tenets of the basic DM theory. Now add to the basic DM theory the following
condition:

[Containment] The premises of an extra-mathematical explanation must contain
all of the information contained within the conclusion and each premise must
contribute some part of that information.

Under this condition, the basic DM theory will classify as explanations all and only those
sound arguments that are classified as explanations by the refined DM theory. In essence, what
the containment condition does is take the relevance constraint and impose it as a condition on
which arguments are explanations, as opposed to a condition on which arguments are valid.
The result, however, is the same. So feel free to kick away the ladder.

7.2 Too Epistemic

Here’s the second objection. According to the refined DM theory, an extra-mathematical
explanation is a sound argument in which all of the information contained within the
conclusion of that argument is contained within the premises. At core, then, the theory makes
use of informational relationships in order to understand explanation. But information, one
might think, is an epistemic phenomenon, and so informational containment has too much of
an epistemic flavour to form the basis of a theory of extra-mathematical explanation.

What exactly is the objection here? The thought, I suppose, is that information is an
epistemic concept, in this sense: information is always information for some knower. Without
knowers and thus without conscious beings, there is no information in the world, and thus no
explanation. To see why this is a problem it is useful to draw a distinction between two very
broad ways of thinking about explanation. Crudely, explanations might be ‘in the world’ or ‘in
the head’. An explanation is ‘in the world’ when it would still exist were there no conscious
beings around to attend to it. An explanation is ‘in the head’, by contrast, when it would not
exist were there no conscious beings around to attend to it.

The objection being raised against the refined DM theory, then, is that it treats explanation
as something that is ‘in the head’. But we all know that explanations are ‘in the world’. At the

28See Williamson ([2013]) for this kind of sentiment.

18



very least, many of the philosophers to whom this paper is speaking — mathematical realists
who motivate their realism by appealing to extra-mathematical explanation — clearly take
explanation to be a worldly phenomenon, and for good reason. If a path to mathematical
realism is to be forged via extra-mathematical explanation, those explanations had better not
be somehow dependent on conscious beings for their existence, on pain of such
mind-dependence infecting the mathematical realism at issue.

There are two responses to this objection. The first is just to take issue with the idea that it
matters at all whether an explanation is ‘in the head’ or ‘in the world’, at least for present
purposes. While it is true that I partially motivated the discussion of the DM theory of
extra-mathematical explanation via the literature on mathematical realism, I did not claim to
be providing a theory of extra-mathematical explanation that may be used by mathematical
realists to defend their view. My goal is the more modest one of showing how to understand
such explanations, on the assumption that there are some. That goal is perfectly compatible
with the resulting theory of extra-mathematical explanation being imbued with an epistemic
flavour, and thus being of limited use to realists.

The second, and potentially more satisfying, response is to deny that information is an
epistemic concept, at least as it is being used in this paper. Consider the proposition that it is
sunny today, in Philadelphia. This proposition seems to carry certain information: it carries
information about the weather, on a particular day in a particular city. Moreover, that the
proposition carries this information appears to be a completely mind-independent
phenomenon. Were there no conscious beings around to attend to the proposition by stating it,
asserting it, knowing it, believing it, and so on, it would still be the case that the proposition
carries the information in question. More generally we may say that there is a bijective
mapping from propositions to information states. I see no reason at all to suppose that this
mapping is an epistemic phenomenon, in the sense described above. There just is a fact of the
matter about the information that is carried by each proposition. And that is all that the refined
DM theory of extra-mathematical explanation ultimately requires. So long as there is a fact of
the matter about the information carried by each proposition that appears in a DM argument,
the explanation that corresponds to the argument in question will be completely independent
of conscious beings in the required sense. Of course, one might think that arguments are some
kind of epistemic phenomenon and so not fit to be explanations. If so, then that is a problem
for all DN-style theories of scientific explanation (in so far as it is a problem at all) and one
that I can reasonably set aside, at least for the time being.

7.3 Informational Containment

The third and final objection against the refined DM theory focuses on the notion of
‘informational containment’ used above. The refined DM theory holds that all of the
information contained within the conclusion of an explanation is contained within the
premises and, moreover, that each premise must contain part of the information contained
within the conclusion. Suppose, then, that we have some claim about the physical world, P,
that we are attempting to explain. Suppose that P is entailed by physical claim Q plus
mathematical claim M such that the resulting sound argument satisfies the tenets of the refined
DM theory. It follows that the mathematical claim M contains part of the information
contained within P. From this it follows that mathematical claims contain information about
the physical universe. But, one might argue, that’s absurd. Surely mathematical claims contain
only information about mathematics; no mathematical claim ever carries physical information!

There are two immediate responses to this objection available. The first response is to just

19



deny that mathematical claims carry only mathematical information. But then the question
arises: how do we make sense of the idea that mathematical claims contain physical
information? I believe we can make sense of this notion by analogy with the laws of nature.
Laws of nature are generalisations that provide high-level structural information about
physical systems. By ‘high-level’ information I mean: information about what a system can
and cannot do. So, for instance, consider the following rough statement of the second law of
thermodynamics: the total entropy of an isolated system always increases over time. This law
dictates what can and cannot happen within an isolated system: it tells us that the total entropy
of an isolated system cannot increase over time; it tells us that the total entropy can decrease
and so on. Call information about what a system can or cannot do: a structural constraint.
Laws of nature often encode information about structural constraints.

Importantly, laws of nature are usually highly idealized, in the following sense: they apply
primarily to ideal systems (see Cartwright [1983]). For instance, consider again the second
law of thermodynamics. Most — if not all — systems in the universe fail to be isolated
systems. So the second law appears to be talking about some ideal system (if you don’t like
the second law of thermodynamics, try Boyle’s ideal gas law). An ideal system, however, is
plausibly an abstract object. The law, then, tells us what is possible or impossible for this
abstract object, at least in the first instance. The law only manages to provide information
about a particular physical system derivatively, via some structural mapping relation. Some
real, physical systems are structurally very similar to the ideal system to which the law
applies. It is in virtue of this structural similarity that the constraints of the ideal system ‘flow
on’ into the real system. That, however, is enough for explanation: because the real system is
held under similar structural constraints as the ideal system, things that are not possible in the
ideal system are not possible in the real system. Of course, the match is not perfect: what we
typically find is that the constraints apply to the real system to some approximation, in virtue
of any structural dissimilarities between the real and ideal system. Sometimes, though, the
match is perfect and we get very precise answers indeed.29

Now, it is not my claim that all laws of nature operate in the manner described above. But
clearly some do. Some laws carry information about the physical universe by (i) carrying
information about an ideal structure first and (ii) by providing information about a physical
system second via a structural mapping between the ideal and physical structures.

Mathematical claims carry information about physical reality in much the same manner.
Some mathematical claims — such as those that are used in extra-mathematical explanations
— also encode high-level structural information. Consider, for instance, the Bridges of
Königsberg. In Königsberg there were once seven bridges connecting a series of islands. It is
impossible to cross all seven bridges of Königsberg, passing over every bridge exactly once.
Why? The explanation lies in graph theory. If we treat each of the seven bridges as an edge,
and each of the islands as a vertex, the seven bridges can be treated as a connected graph. The
resulting graph is non-Eulerian, which means that it provably lacks an Eulerian path. An
Eulerian path is just a continuous path through a graph that passes over each edge exactly
once. The lack of Eulerian paths explains why no-one can cross the seven bridges in the
manner described. In this case, mathematical claims about non-Eulerian graphs encode
structural information about the kinds of paths that can or cannot obtain through certain
mathematical structures. In the first instance, then, these mathematical claims don’t yet
provide information about any physical system. However, once various structural mappings

29For more on the different kinds of structural mapping relations that are available see
Bueno and Colyvan ([2011], pp. 348–350).

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are implemented between the mathematical structure and the physical system then
mathematical claims come to provide information about the physical system in question,
albeit indirectly. The picture, though, is the same as in the nomic case. Mathematical claims,
like some laws, provide information about some abstract object in the first instance, and then
they provide information about a physical object via a structural mapping between the abstract
and physical objects. Note that since structural mappings are, themselves, mathematical
objects, at least some mathematical objects carry information about the physical world
directly. But this is nothing new, as it seems to be implied in the nomic case anyway.

The preceding discussion leans heavily on Pincock’s ([2004, 2007]) structural mapping
account of applied mathematics. As Bueno and Colyvan argue, however, Pincock’s account
leaves little room for mathematics to play an explanatory role in science (Bueno and Colyvan
[2011], p. 351). Structural mapping is a form of representation and, as such, a form of
description. Accordingly, if the only information that mathematics carries is descriptive
information regarding physical structures, then my account of informational containment runs
the risk of conflating the descriptive and explanatory roles of mathematics.

The problem may be avoided by appealing to Bueno and Colyvan’s inferential conception
of applied mathematics. According to the inferential conception, the application of
mathematics proceeds in three stages: (i) immersion, whereby a physical system is mapped
into a mathematical structure, (ii) inference, whereby inferences are drawn within
mathematics about the relevant mathematical structure, and (iii) interpretation, whereby the
results of the inferential step are interpreted physically. Note that results are read back down
into the physical system via some structural mapping relation between the mathematical and
physical structures. Importantly, the structural mapping used in the interpretation step need
not be the inversion of the mapping used at the immersion step.

Bueno and Colyvan ([2011], p. 366) maintain that the inferential conception is better suited
to handling mathematical explanation than is the structural mapping conception. Although,
according to the inferential conception, mathematics is used to describe a physical system via
a structural mapping, the mathematics does more than merely describe. Claims about
mathematical structures are deduced, and then these claims are used to provide non-descriptive
information about physical systems via the interpretation step. Structural mapping is a bridge
across which non-descriptive information regarding a physical system may be conveyed.

Bueno and Colyvan’s inferential conception therefore provides a useful framework for
elucidating the difference between descriptive and non-descriptive physical information (a
distinction that was used in my response to the genuineness problem in Section Six). M
contains descriptive physical information relevant to P when M contains information about an
aspect of some mathematical structure that is mapped into the physical structure
corresponding to P. M contains non-descriptive physical information relevant to P when M
contains information about the physical structure corresponding to P and it does so in virtue of
carrying information about an aspect of a mathematical structure and yet that aspect is not a
relatum in a mapping relation between the mathematical and physical structures at issue.
Importantly, information about what is possible or impossible within a mathematical structure
is non-descriptive information; mapping relations relate structures, they don’t relate modal
facts about structures. Nonetheless, modal facts about one structure can be deduced about
another structure in virtue of a mapping. I take this to be one of the central insights of the
inferential dimension of Bueno and Colyvan’s proposal: there is a difference between the
aspects of a mathematical structure that can be mapped to a physical structure, and the aspects
of a mathematical structure that cannot be mapped but that nonetheless provide information
about a physical structure (in this case, in virtue of a mapping relation).

21



In sum, then, mathematical claims, like laws, carry information about physical systems, and
they convey this information via structural mappings. The structural mappings do not,
however, exhaust the informational content of the relevant mathematical claims. Rather, the
structural mappings constitute a point of contact across which non-descriptive information
may be conveyed. The information in question being modal information about what a physical
system can or cannot do.

Of course, the account is still limited in so far as it won’t extend to cases in which
mathematics plays an explanatory role in science despite there being no structural mappings in
place. As Batterman ([2010]) has urged, however, there are some such cases. For these cases,
we might appeal to Batterman’s own account of applied mathematics. On this account,
mathematics carries information about physical systems indirectly, by carrying information
about physical regularities. Mathematics, very roughly, is used to explain why the laws are
laws. Mathematics plays this role by carrying information about the limits of regularities:

[M]any applied mathematical accounts of the robustness of patterns [...] involve
the investigation of singularities. To explain and understand the robustness of
patterns and regularities, one sometimes needs to focus on places where those
very regularities break down. Thus, the existence of patterns or regularities in the
world, and our desire to understand and explain them, has led rather
straightforwardly to investigate singularities in mathematical limiting operations.
(Batterman [2010], p. 21)

A singularity typically arises when a mathematical object is not defined. One way for this to
happen — and the case that interests Batterman — is when the limit of a function is both
unbounded and the behaviour of the function is different in the infinite limit compared to the
finite progression of the function toward the limit. When such singularities are involved, the
mathematics of asymptotic operations carries information about a physical regularity.
Mathematics carries this information despite the absence of any structural mappings between
the mathematics of asymptotic operations and a physical system. It does this because the
mathematics limns the boundaries of the physical regularity directly. Batterman’s approach is
compatible with the broad approach to informational containment sketched above. According
to that account, mathematical claims carry information about physical systems in the same
way that laws do. Many laws, however, carry information about why it is that other laws hold
and on Batterman’s view, that is exactly what mathematics does.30

As noted, there are two responses to the broad worry about informational containment
under discussion. The second response is to deny that mathematical claims ever carry physical
information about physical systems. Instead, the reverse is true: claims about physical systems
(sometimes) carry mathematical information. Here’s the basic idea: it is not just mathematical
objects such as numbers, sets, functions, and so on that possess mathematical properties.
Rather, some physical objects also possess mathematical properties. This might seem like a
strange idea at first, but examples can be given that can help us to better grasp the notion. Here
are four such examples. First example: spacetime is thought to be a physical object. But it is
also thought that spacetime is continuous.31 Continuity, however, is a mathematical property.
So spacetimes have mathematical properties. Second example: physical objects possess

30As Steiner ([1995], p. 137) notes, this view was defended by Frege.
31Spacetime is assumed to be continuous within the general theory of relativity (see

Einstein ([2010]). Recent developments in quantum gravity suggest that spacetime may be
discrete, however. This is so for (at least) string theory and causal set theory.

22



shapes. The property of having a certain shape, however, is a mathematical property,
enshrined in geometry and describable algebraically. So physical objects have mathematical
properties. Third example: some life-cycles — such as the cycling of the cicada life-cycle
length — possess periodicity. Periodicity, however, is a mathematical property, shared by unit
cycles more generally, which are abstract objects that may be described within pure
mathematics. So some life-cycles have mathematical properties. Fourth example: spacetime
has the property of being curved. Curvature, however, is a mathematical property, one that
features heavily in the Riemannian geometry that underlies the metric interpretation of general
relativity. So spacetimes have mathematical properties.

I don’t wish to defend these examples here; they’re really just supposed to motivate a much
grander thesis about the relationship between the mathematical and the physical. Suppose,
however, that the examples are apt and physical systems really do possess mathematical
properties. Then it is no surprise that mathematical claims carry information about physical
systems, despite not carrying any physical information whatsoever. An extra-mathematical
explanation is just a way of bringing the relevant physical and mathematical information
together in order to explain some physical-cum-mathematical complex; namely, a complex
object consisting of physical and mathematical properties.

The idea that physical systems possess mathematical properties dovetails nicely with
Davide Rizza’s ([2013]) recent account of applied mathematics. The point of departure for
Rizza’s account, like Batterman’s, is the observation that not all cases of applied mathematics
involve structural mappings, and so the structural mapping and inferential conceptions of
applied mathematics fail to be fully general. According to Rizza, some cases of applied
mathematics operate as follows. First, one identifies some formal property of a physical
system. Having identified a formal property of the physical system, one then reasons
mathematically about it. These inferences are then used to derive further results regarding the
physical system. In other words, on Rizza’s view the inferences that are used in the inferential
conception of applied mathematics can be conducted in cases where there are no structural
mappings. Rather, the target of the inferences are the formal properties that physical systems
actually possess. Rizza’s account of applied mathematics therefore involves the idea that
physical systems sometimes possess mathematical properties, which makes it a useful model
for the account of informational containment being proposed.

Now, I recognize that the two responses to the worry about informational containment
outlined in this section may inspire more controversy than I have space to fully engage with.
For one thing, both responses seem to have a distinctively realist flavour about them: we either
need abstract structures or mathematical properties. It pays, then, to offer a third response to
the worry. Here it is: providing an account of how it is that mathematical claims contain
physical information is everyone’s problem; at least, everyone who believes that there are
extra-mathematical explanations. In asserting that mathematical claims explain physical
claims, it is difficult to see how one can thereby withold commitment from the view that
mathematical claims contain information about physical systems. For suppose that
mathematical claims contain no such information at all. Then it seems utterly mysterious as to
how mathematical claims could ever come to explain physical claims. After all, if
mathematical claims contain no information about physical systems then it doesn’t seem much
of a stretch to say that mathematical claims are incapable of telling us anything about physical
systems whatsoever; the two ‘realms’ appear to be completely distinct. So telling a plausible
story about how mathematical claims contain information about physical systems is a rather
general concern that any theory of extra-mathematical explanation must ultimately face.
Which is just to say that the question of how mathematical claims carry physical information

23



is really a general question in applied mathematics, and so will need to be taken up in that
context.32 I have, however, gestured toward some of the ways in which this might be done, by
indicating where my account of informational containment may link up with various theories
of mathematical application.

8 Conclusion

Here, at last, is a statement of the DM theory, with all the bells and whistles:

The Full-Blown DM Theory of Extra-Mathematical Explanation

1. Extra-mathematical explanations are sound R-Arguments, where an R-Argument is an
argument in which all of the information contained within the conclusion of the
argument is contained within the premises, and each premise contributes some part of
the information contained within the conclusion.

2. The conclusion of an extra-mathematical explanation is a proposition stating the
physical phenomenon to be explained.

3. Among the premises of an extra-mathematical explanation there must be at least one
mathematical claim.

4. An extra-mathematical explanation must satisfy the Genuineness Condition.

5. Mathematical claims contain information about physical systems either indirectly, via
structural mappings, or directly in virtue of physical objects possessing mathematical
properties.

The full-blown DM theory is modeled closely on early DN accounts. The theory will
require further refinement to bring it into line with contemporary versions of the DN theory. In
fact, achieving such alignment appears quite promising, at least with respect to Jansson’s
(forthcoming) DN theory. On this theory, the subsumption of an explanandum under a law is
understood in terms of inference tickets: inferential licenses that enable one to infer some
explanandum from a group of claims that includes a law. According to Restall’s (1996)
channel theory, the informational containment of ∆ by Γ ultimately amounts to the existence
of a license to infer the information in ∆ from the information in Γ, where the inferential
licenses themselves are given by the manner in which laws, metaphysical relationships,
conventions, and so on control the flow of semantic information. The refined DM theory may
therefore be combined with Jansson’s version of the DN theory by adopting this
channel-theoretic account of the relevance relation, using it as a foundation for explanation.

Acknowledgements

I am indebted to Alan Baker, Mark Colyvan, Aidan Lyon, Daniel Nolan, Andrew Tedder, and
David Ripley for useful discussion of this paper. I am also indebted to the University of
Connecticut for hosting me as a visitor during the preparation of this work. Versions of this
paper were presented at the Explananza Workshop at the University of Western Australia in

32See Steiner ([1995]) for an excellent overview of the problems of applied mathematics
along with a discussion of the view that mathematical facts are the ‘laws of the laws of nature’.

24



2015 and the Australasian Association of Philosophy Conference in 2015. Research on this
article was partly supported by a Humanitites Travelling Fellowship from the Australian
Academy of the Humanities (HTF201500015).

Sam Baron
University of Western Australia

Philosophy
M204 School of Humanities

35 Stirling Highway, Crawley 6009
samuel.baron@uwa.edu.au

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